Subgroups — C₂×C₅⋊(C₅⋊(D₆₁⋊C₃)) #32913076

Order 18300.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₆₁⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅⋊(C₅⋊(D₆₁⋊C₃))	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅⋊D₃₀₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅²×C₆₁⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅⋊(D₆₁⋊C₃)	—	5	1	no	maximal
6	C₂×C₅⋊(D₆₁⋊C₃)	—	5	5	no	maximal
7	C₅⋊D₃₀₅	C₂×C₃	1	2	yes	complemented
8	C₂×C₅²×C₆₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×C₅×C₆₁⋊C₃	D₅	1	1	yes	characteristic, complemented
10	C₂×C₅×C₆₁⋊C₃	D₅	1	5	yes	complemented
11	C₅⋊(D₆₁⋊C₃)	—	5	2	no	—
12	C₅⋊(D₆₁⋊C₃)	—	5	10	no	—
13	C₅²×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
14	C₂×D₃₀₅	—	5	1	no	—
15	C₂×D₃₀₅	—	5	5	no	—
16	C₅×C₆₁⋊C₃	C₂×D₅	1	1	yes	characteristic, complemented
17	C₅×C₆₁⋊C₃	C₂×D₅	1	5	yes	complemented
18	C₂×D₆₁⋊C₃	—	25	1	no	Hall
19	C₂×C₅×C₆₁	C₃×D₅	1	1	yes	characteristic, complemented
20	C₂×C₅×C₆₁	C₃×D₅	1	5	yes	complemented
21	D₃₀₅	—	5	2	no	—
22	D₃₀₅	—	5	10	no	—
23	C₂×C₆₁⋊C₃	C₅⋊D₅	1	1	yes	characteristic, complemented
24	D₆₁⋊C₃	—	25	2	no	—
25	C₅×C₆₁	C₂×C₃×D₅	1	1	yes	characteristic, complemented
26	C₅×C₆₁	C₂×C₃×D₅	1	5	yes	complemented
27	C₂×C₃×C₅⋊D₅	—	61	1	no	maximal, Hall
28	C₂×D₆₁	—	25	1	no	Hall
29	C₆₁⋊C₃	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₅²	—	122	1	no	—
31	C₃×C₅⋊D₅	—	61	2	no	—
32	C₂×C₆₁	C₃×C₅⋊D₅	1	1	yes	characteristic, complemented
33	D₆₁	—	25	2	no	—
34	C₂×C₅⋊D₅	—	61	1	no	Hall
35	C₃×C₅²	—	122	1	no	Hall
36	C₆₁	C₂×C₃×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃×D₅	—	305	1	no	—
38	C₂×C₃×D₅	—	305	5	no	—
39	C₂×C₅²	D₆₁⋊C₃	1	1	yes	characteristic, complemented
40	C₅⋊D₅	—	61	2	no	—
41	C₂×C₃×C₅	—	61	1	no	—
42	C₂×C₃×C₅	—	122	5	no	—
43	C₃×D₅	—	305	2	no	—
44	C₃×D₅	—	305	10	no	—
45	C₅²	C₂×D₆₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
46	C₂×D₅	—	305	1	no	—
47	C₂×D₅	—	305	5	no	—
48	C₃×C₅	—	61	1	no	—
49	C₃×C₅	—	122	5	no	—
50	C₂²×C₃	—	1525	1	no	Hall
51	C₂×C₅	C₅⋊(D₆₁⋊C₃)	1	1	yes	characteristic, complemented
52	C₂×C₅	C₅⋊(D₆₁⋊C₃)	1	5	yes	complemented
53	D₅	—	305	2	no	—
54	D₅	—	305	10	no	—
55	C₂×C₃	—	122	1	no	—
56	C₂×C₃	—	1525	2	no	—
57	C₅	C₂×C₅⋊(D₆₁⋊C₃)	1	1	yes	characteristic, complemented
58	C₅	C₂×C₅⋊(D₆₁⋊C₃)	1	5	yes	complemented
59	C₂²	—	1525	1	no	Sylow
60	C₃	—	122	1	no	Sylow
61	C₂	—	1525	2	no	—
62	C₂	C₅⋊(C₅⋊(D₆₁⋊C₃))	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₅⋊(C₅⋊(D₆₁⋊C₃))
commutator, cosocle, hypocenter	C₅²×C₆₁	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₆₁	C₂×C₃
LCS	G ▷ C₅²×C₆₁
UCS	C₁ ◁ C₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₅²×C₆₁⋊C₃	C₂	semidirect
2. C₅⋊(C₅⋊(D₆₁⋊C₃))	C₂	direct (N × H)
3. C₂×C₅⋊D₃₀₅	C₃	semidirect
4. C₅²×C₆₁⋊C₃	C₂²	semidirect
7. C₅⋊D₃₀₅	C₂×C₃	semidirect
8. C₂×C₅²×C₆₁	C₂×C₃	semidirect
9. C₂×C₅×C₆₁⋊C₃	D₅	semidirect
10. C₂×C₅×C₆₁⋊C₃	D₅	semidirect
13. C₅²×C₆₁	C₂²×C₃	semidirect
16. C₅×C₆₁⋊C₃	C₂×D₅	semidirect
17. C₅×C₆₁⋊C₃	C₂×D₅	semidirect
19. C₂×C₅×C₆₁	C₃×D₅	semidirect
20. C₂×C₅×C₆₁	C₃×D₅	semidirect
23. C₂×C₆₁⋊C₃	C₅⋊D₅	semidirect
25. C₅×C₆₁	C₂×C₃×D₅	semidirect
26. C₅×C₆₁	C₂×C₃×D₅	semidirect
29. C₆₁⋊C₃	C₂×C₅⋊D₅	semidirect
32. C₂×C₆₁	C₃×C₅⋊D₅	semidirect
36. C₆₁	C₂×C₃×C₅⋊D₅	semidirect
39. C₂×C₅²	D₆₁⋊C₃	semidirect
45. C₅²	C₂×D₆₁⋊C₃	semidirect
51. C₂×C₅	C₅⋊(D₆₁⋊C₃)	semidirect
52. C₂×C₅	C₅⋊(D₆₁⋊C₃)	semidirect
57. C₅	C₂×C₅⋊(D₆₁⋊C₃)	semidirect
58. C₅	C₂×C₅⋊(D₆₁⋊C₃)	semidirect
62. C₂	C₅⋊(C₅⋊(D₆₁⋊C₃))	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅²×C₆₁⋊C₃	2
C₅⋊(C₅⋊(D₆₁⋊C₃))	2
C₂×C₅⋊D₃₀₅	3
C₂×C₅⋊(D₆₁⋊C₃)	5
C₂×C₃×C₅⋊D₅	61

Subgroups — C2×C5⋊(C5⋊(D61⋊C3)) #32913076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅⋊(C₅⋊(D₆₁⋊C₃)) #32913076